Hybrid design of PID controller for four DoF lower limb exoskeleton

Applied Mathematical Modelling 72 (2019) 17–27

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Hybrid design of PID controller for four DoF lower limb exoskeleton Mohammad Soleimani Amiri a, Rizauddin Ramli a,∗, Mohd Faisal Ibrahim b a

Centre for Materials Engineering and Smart Manufacturing, Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, Malaysia b Center for Integrated Systems Engineering and Advanced Technologies, Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, Malaysia

a r t i c l e

i n f o

Article history: Received 2 August 2018 Revised 1 March 2019 Accepted 5 March 2019 Available online 13 March 2019 Keywords: Lower limb exoskeleton Proportional-integral-derivative Genetic algorithm Particle Swarm Optimization

a b s t r a c t In this paper, a method of tuning a proportional-integral-derivative controller for a four degree-of-freedom lower limb exoskeleton using hybrid of genetic algorithm and particle swarm optimization is presented. Transfer function of each link of the lower limb exoskeleton acquired from a pendulum model, was used in a closed-loop proportional-integralderivative control system, while each link was assumed as one degree-of-freedom linkage. In the control system, the hybrid algorithm was applied to acquire the parameters of the controller for each joint for minimizing the error. The algorithm started with genetic algorithm and continued via particle swarm optimization. Furthermore, a 3-dimensional model of the lower limb exoskeleton was simulated to validate the proposed controller. The trajectory of the control system with optimized proportional-integral-derivative controller via hybrid precisely follows the input signal of the desired. The result of the hybrid optimized controller was compared with genetic algorithm and particle swarm optimization based on statistics. The average error of the proposed algorithm showed the optimized results in comparison with genetic algorithm and particle swarm optimization. Furthermore, the advantages of the hybrid algorithm have been indicated by numerical analysis. © 2019 Elsevier Inc. All rights reserved.

1. Introduction In recent decades, as aging population and individuals suffering from mobile disability have been increased, the requirements for exoskeleton have been more significant [1,2]. Hence, researchers have paid more attentions to design an exoskeleton as rehabilitation devices. The exoskeleton is a kind of wearable robot employed to improve the physical stamina of the human’s muscle. Furthermore, the devices assist patients who have lost their abilities to carry out the tasks of daily living because of diseases such as strokes, accidents and spinal cord injuries (SCI). Lower Limb Exoskeleton (LLE) which works in parallel with human limb is used in purpose of rehabilitation to bring back the physical and mental capabilities of patients who cannot walk [3,4]. The LLE is also functioned as a gait training robot which requires a controller to reduce the steady state errors and improve the stability of the robot. Different methods and algorithms of controlling the LLE have been studied in literatures [5,6]. Hussain et al. [7] examined an adaptive impedance control to provide interactive robotic gait training. This control ∗

Corresponding author. E-mail address: [email protected] (R. Ramli).

https://doi.org/10.1016/j.apm.2019.03.002 0307-904X/© 2019 Elsevier Inc. All rights reserved.

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method can learn the disability level and effort of the patient in real time. On the other study, Wu et al. [8] designed a six Degree of Freedom (DoF) hip exoskeleton and developed real-time control system based on Proportional-Integral-Derivative (PID) controller in passive control mode to perform trajectories tracking task. In addition, a Fuzzy Logic Controller (FLC) was also applied in active mode to perform walking assistant task. He et al. [9] developed an adaptive neural network controller for a two DoF simulated LLE model without the knowledge of the dynamic model of it. An adaptable robust controller was developed by Wu et al. [10] for a three DoF lower limb rehabilitation robot (LLRR) to overwhelm the system uncertainties and overcome patients’ disturbance. The results were validated by simulation and experiments which show each joint followed its desired training trajectory. Han et al. [11] studied a control system for a 12 DoF lower limb multifunctional exoskeleton. Their control system was a model-free based adaptive nonsingular fast terminal sliding mode control using intelligent Proportional-Integral (PI) controller to obtain fast convergence in tracking error. The performance of the control system intensely depends on the controller, hence the process of tuning has major roles in its operation. Elbayomy et al. [12] discussed a PID controller using Genetic Algorithm (GA) for the Electro-Hydraulic Servo Actuator System (EHSAS) to obtain the effective results in comparison with classical PID controller. Similarly in the other work, PID controller was tuned by coupling Gain Phase Margin (GMP) method with GA for servo control system [13]. Mohanty et al. [14] presented gain tuning of PID controller using Differential Evolution (DE) optimization method for a multi-unit source power system. Azar et al. [15] used a Neural Network (NN) for controlling a 6-bar Steward parallel robot for rehabilitation purpose. They applied GA to optimize the controller’s parameters. For the optimization of LLE controller, hybrid method which is a combination of two or more methods has been used to increase the chance of achieving more optimal results [16,17]. Farand et al. [18] developed hybrid method of GA, Particle Swarm Optimization(PSO), and Symbiotic Organism Search (SOS) to increase the computational time and accuracy of their proposed method in comparison with other known methods such as GA, DE, PSO for high dimensional and complex function. Shunmugapriya et al. [19] used a hybrid method based on the characteristic of Ant Colony Optimization (ACO) and Artificial Bee colony (ABC) for feature selection and classification to decrease the size of feature subset and increase the accuracy of classification. Mart-nez-Soto et al. [20] studied a hybrid method using GA and PSO to optimize the FLC and verified their method on autonomous mobile robot. In this paper, our main contribution is to employ a hybrid of GA and PSO (GaPSo) for tuning the PID controller of LLE. The motivation of GaPSo is to minimize the trajectory error of the LLE for different rehabilitation gait training that is used by physiotherapists. This paper presents a method of tuning the PID controller’s gains optimally by using GaPSo. The transfer function for each link of the LLE was acquired from a pendulum model which subsequently used in a closed-loop PID controller. In the control system, GaPSo is utilized to determine the optimized PID parameters by minimizing the error of the system. In the proposed algorithm, firstly, GA was run with random initial value then, the results of the GA created a search space for PSO to find the optimum PID parameters for the control system. The optimized controller is implemented in the closedloop control system with step response and in a 3 Dimensional (3D) model of the LLE in Gazebo environment. In addition, the result is compared with the GA and PSO for two different conditions of Range Of Motion(ROM), in which one joint is actuated while the other joints are locked, and also in gait training while all the joints move simultaneously. 2. Kinematic model Fig. 1 illustrates the free body diagram in saggital plane of one leg of LLE in which one DoF is attached at revolute joint of hip and another DoF is fixed at the knee. O1 and O2 are hip and knee joints, which are revolute joints and the O3 is the fixed ankle joint. In addition, l1 , l2 , lG1 , and lG2 represent the length of femur and tibia and their Center of Gravity (CoG), respectively. Several methods such as Lagrangian, Newton–Euler, Kane, and Hamilton equations are common techniques that have been adapted to determine dynamic equations of robot manipulators [10,21]. In this paper, dynamic modeling of the LLE is determined to establish the relationship between the forces of the joints’ actuator and the generated movement. Each link of the LLE is assumed as one DoF pendulum model, as shown in Fig. 2, to determine the dynamic equation. On the other words, while one joint rotates freely, the other joint is fixed. The equation of motion is expressed as the summation of torques at point Oi , where i = 1, 2, and given by,

Ti = (Mi + mi )glGi sin(θi ) + T fi + Ii θ¨i

(1)

where, Mi is denoted as the influence of the mass and inertia of the fixed link. mi , Ti , g and T fi represents the mass of the link, external torque, gravitational acceleration and viscous friction respectively. In addition, Ii and lGi and θ¨i are the mass moment of inertia, length of CoG, and angular acceleration, respectively.

Mi = mi+1 + mi+2

(2)

Ii = Iyy + Mi li2

(3)

lGi =

Mi li + 0 .5 mi li Mi + mi

(4)

M.S. Amiri, R. Ramli and M.F. Ibrahim / Applied Mathematical Modelling 72 (2019) 17–27

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Fig. 1. Free body diagram of the LLE.

Fig. 2. Free body diagram of the pendulum model.

T fi = bi θ˙ i

(5)

where, bi is the friction coefficient. Solving Eqs. (1)–(5),the torque is determined as,

Ti = θ¨i +

bi ˙ (M + mi )glGi θi + i θi Ii Ii

(6)

Next, Laplace function of Eq. (6) is solved as,

Ti (s ) = θi (s )s2 +

bi (M + mi )glGi θi ( s ) s + i θi ( s ) Ii Ii

(7)

The plant transfer function is derived as,

Gi ( s ) =

θi ( s ) 1 = i )glGi Ti (s ) s2 + bIii s + (Mi +m Ii

(8)

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M.S. Amiri, R. Ramli and M.F. Ibrahim / Applied Mathematical Modelling 72 (2019) 17–27 Table 1 Features of the links.

mass (Kg) length (m) inertia (kg m2 )

femur

tibia

foot

0.2943 0.40 5.7 × 10− 3

0.2159 0.40 8.6 × 10− 5

0.11504 – –

Fig. 3. Block diagram of the control system.

Fig. 4. Block diagram of the PID controller.

Table 1 depicts physical features of mass and length for femur, tibia, and foot. Based on the table, the transfer functions of one leg of the LLE which consists of femur G1 and tibia G2 are shown as follows,

G1 ( s ) =

θ1 ( s ) 1 = 2 T1 (s ) s + 15.33s + 31.96

(9)

G2 ( s ) =

θ2 ( s ) 1 = 2 T2 (s ) s + 48.67s + 47.32

(10)

For the computational purpose of the transfer function of the femur, it is assumed that the knee joint is fixed, and similarly, for obtaining the transfer function of the tibia, the hip is assumed as a fixed joint. 3. Trajectory error The closed-loop control system of the LLE is represented in Fig. 3. The input and output to the control system are the desired and actual trajectory angle of the hip and knee joint respectively. A PID controller is used for controlling the joints of the LLE, due to its potential in providing satisfactory results and comfort of operation [22]. Closed-loop transfer function of the control system is represented as,

θact (s ) C ( s )Gi ( s ) = θdes (s ) 1 − C (s )Gi (s )

(11)

where, C(s) is the transfer function of the controller, and Gi (s) is the transfer function of femur when the i = 1 and it is tibia’s transfer function when i = 2. The PID controller is shown in Fig. 4. The error e(s) is difference between the actual and desired trajectory of each joint [23], which is expressed by the following equation,

e(s ) = θdes (s ) − θact (s )

(12)

where, θ act (s) and θ des (s) represent the actual and desired trajectory angle of the LLEs’ joints respectively. The transfer function of PID controller is written as follows,

C (s ) =

T (s ) Ki = Kp + + Kd s e (s ) s

(13)

M.S. Amiri, R. Ramli and M.F. Ibrahim / Applied Mathematical Modelling 72 (2019) 17–27

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Fig. 5. Configuration of the initial population.

where, Kp , Ki , and Kd are the gain of proportional, integral, derivative actuator respectively. Error, e(s), is denoted as the input to the PID controller. T(s) is the output of the controller, which is summation of errors influenced by the proportional, integral, and derivative actions. In order to determine the objective function of the optimization problem, the input of the control system is fixed as a step response (θdes = 1 ). Therefore, by substitution of Eqs. (11) in (12), the error, eobj (s), is represented as,

eob j (s ) = 1 −

C ( s )Gi ( s ) 1 − C ( s )Gi ( s )

(14)

The objective function is defined based on Integral Time Absolute Error (ITAE) [24], which is the summation of absolute error, weighted by time and sample time. The objective function is shown by the following equation,



fob j =

∞ 0

t |eob j (t )|dt

(15)

where, t is the elapsed time; eobj (t) is the error of the control system in time domain. 4. Tuning of PID controller In this paper, tuning of PID controller is defined as optimization problem and solved by the combination of GA and PSO in series, known as GaPSo. First, the optimization is carried out by GA and the optimized Kp , Ki , and Kd values is used as initial values of PSO. In this case, GA provides search space for PSO in the neighbor of the global optima, then PSO converges to the optimum point. GA is based on the concept of natural selection and genetics, which finds the values of the process in the way that the best output be obtained [25]. Meanwhile, PSO is a population-based optimization strategy, which is inspired by biological behavior of swarms, birds and fish school [26]. PSO resembles GA in terms of setting initial population, and finding optimum solution iteratively, although, there is no mutation and crossover. PSO has been implemented in various types of problem and proven to converge faster than GA in achieving global optimal solution, but easy to trap in local optima [27]. The design variables of optimization problem are three parameters of PID controller. Fig. 5 illustrates the configuration of the first population, in which each gene consists of 30 set of random values for PID parameters. After creation of the initial population, the objective function is determined to evaluate each gene. Then the genes are arranged based on the value of the objective function by sorting from the lowest the highest. The generation are created by crossover, mutation and some of them, which evaluated as the best gene, remains unchanged. These genes are called “elite genes”. Crossover extract the genes from the population and recombine them to increase the chance in finding the better result. Mutation remains the diversity of GA from one generation to the next one. The process of creating the next generation consists of keeping 5 percent of the previous population unchanged as elite genes. These genes should have the best results after evaluation. The probabilities of crossover and mutation are 0.8 and 0.2. It means that, 80 percent of the remain population are selected via crossover. Finally, the rest be filled using mutation. The Guassian distribution is selected for mutation [28]. Therefore, out of 30 genes in each population, two of them are elite genes and remained unchanged from one generation to the next one; 22 genes are filled by crossover; and six genes are selected via mutation. The evaluation of each gene is repeated for the next generation. The cycle of evaluation continues until the iteration of GA terminated. Eq. (16) shows the output of the GA.

xga = [K p Ki Kd ]

(16)

The further iterations continues by PSO, which includes particles. As shown in Eq. (17), each particle carries the PID parameter values,

xi, j = [K p Ki Kd ]i j

(17)

where, i and j are the number of iteration and population respectively. The first particles of PSO are selected randomly based on the output of PID parameters resulted from GA, as shown in following equation,

x1, j = rand ([xga −  , xga +  ] )

(18)

where,  is defined as parameter tolerance. The population of the next iteration is created as follows,

xi, j = xi−1, j + vi, j

(19)

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M.S. Amiri, R. Ramli and M.F. Ibrahim / Applied Mathematical Modelling 72 (2019) 17–27 Table 2 Optimized PID parameters.

hip knee

Kp

Ki

Kd

78.5522 148.0216

188.6406 106.1593

1.2536 0.0059

where, vi, j represents the velocity and direction of each particle toward the particle of the next iteration; xi−1, j is the particle of previous iteration. Equation of velocity is shown as follows,

vi, j = ωi × vi−1, j + c1 × rand1 × pbest,i−1 − c2 × rand2 × gbest − xi, j

(20)

where, rand1 and rand2 are the random values which are between 0 and 1; c1 and c2 are positive coefficients of the selfrecognition component, and social components respectively,which are normally set as 2; pbest,i and gbest are defined as best position of each population and global best of them respectively; ωi is called as the inertia weight where its value readjusted as the following equation per each iteration,

ωi = ωdamp × ωi−1

(21)

where, ωdamp is set as 0.05. In each iteration the objective function is determined to evaluate the particles. Each particle that has minimum value of objective function is stored as pbest,i . Among the stored value of pbest,i the lowest value is selected as global best, which shown as gbest . The particle of gbest will be chosen as the final result of the GaPSo.

f or

i = 1, 2, . . . , 150

pbest,i = min{ f (xi, j )}

j = 1, 2, . . . , 30

gbest = min{ pbest,i }

(22)

Fig. 6 shows the flow chart of GaPSo. Algorithm 1 exhibits the pseudo code of the GaPSo. Algorithm 1 Hybrid algorithm of GaPSo. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22:

Start GA initialize random population with 30 genes; Evaluate initial population; while Number of generation less than 300 do; create new generation using crossover and mutation; Evaluate the population Sort the population end while End GA; Start PSO; initialize population with 30 particles around the output of GA results; Evaluate initial population; Selecting the initial pbest while Number of iteration less than 150 do; create new iteration; Select the pbest ; if pbest less than the previous gbest then; store it as gbest end if end while select the particle of gbest of the last generation as result End PSO

In this paper, the maximum iteration of the GaPSo is set 450, in which 300 iterations are done by GA and the rest continues via PSO. The population of both GA and PSO is 30. Table 2 illustrates the results of the GaPSo. Fig. 7 represents the value of the best objective function in each iteration of the GaPSo. The value of best objective function, determined via GA, is converged gradually to the global optima. However, the objective function converged asymptotically during the PSO, running at the last one-third of total iterations.

M.S. Amiri, R. Ramli and M.F. Ibrahim / Applied Mathematical Modelling 72 (2019) 17–27

Fig. 6. Flow chart of the GaPSo.

(a) Hip

(b) Knee

Fig. 7. The best objective function value in each populations.

23

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M.S. Amiri, R. Ramli and M.F. Ibrahim / Applied Mathematical Modelling 72 (2019) 17–27

(b) Knee

(a) Hip Fig. 8. Step response of controller for each joint.

Fig. 9. 3D model of LLE in Gazebo.

5. Results In order to verify the performance, GaPSo has been added to the simulation model of the control system, as shown in Fig. 3. The optimized PID parameters, which are Kp , Ki , and Kd are adjusted according to trajectory performance as shown in Table 2. The results of this verification with controller are shown in Fig. 8, where the desired trajectory is a step response and GaPSo is the actual trajectory. In addition, control system by GA and PSO are simulated and their results are compared with GaPSo. Fig. 8a and b are the step response of controller for hip and knee, respectively. In Fig. 8a the rise time for GaPSo is less than GA and PSO by 0.1249 and 1.3139 s, respectively. Similarly, in Fig. 8b, rise time for GaPSo is 2.133 and 5.578 seconds less than GA and PSO, respectively. The optimized PID controller was developed in 3D model of the LLE to simulate its behavior in real environment. Fig. 9 illustrates the model animated and described in Gazebo platform, in which the optimized PID controller was implemented for the hip and knee of the both legs. Gazebo is a 3D open source dynamic simulator, and it is one of the platforms of the Robot Operating System (ROS) [29]. The physical features of the LLE model resembled the mass and length of the pendulum model. The 3D model was tested to validate the performance of the GaPSo in two different experiments, which are ROM and gait training. In ROM training, the performance of each joint is observed, while the other joints are locked [10,30,31]. Fig. 10 compares the trajectory performance of four joints for GaPSo, GA, and PSO.

M.S. Amiri, R. Ramli and M.F. Ibrahim / Applied Mathematical Modelling 72 (2019) 17–27

(a) Left hip

(b) Right hip

(c) Left knee

(d) Right knee

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Fig. 10. Trajectory angle for each joint under ROM condition. Table 3 Error of trajectories for each joint. Right joints

EGA

EPSO

EGaPSo

P − value

Hip Knee

0.0062 0.0073

0.0359 0.0288

0.0043 0.0045

2.4214E-10 1.2677E-5

Left joints Hip Knee

EGA 0.0067 0.0049

EPSO 0.0360 0.0231

EGaPSo 0.0043 0.0034

P − value 2.5116E-10 4.7582E-7

Table 4 Error of trajectory for GaPSo. Joint

Hip Knee

Left

Right 2

ME

AE

RMS

R

0.0379 0.0398

0.0072 0.0111

0.0094 0.01651

0.9974 0.9980

ME

AE

RMS

R2

0.0182 0.0427

0.0073 0.0146

0.0094 0.0170

0.9974 0.9980

In the gait training condition, all the joints moved simultaneously to test the performance of the LLE model. The actual trajectory of the 3D model of the LLE with optimized PID controller via GaPSo, GA, and PSO is compared in this Fig. 11. Table 3 illustrates the average error of trajectory for each joint. EGA , EPSO , and EGaPSo are the average error of 10 times running under gait trainings with various PID parameters. P − value comes from the ANOVA test for the error of ten different running. In Figs. 10 and 11, the results show that GaPSo optimized the PID parameters of each joint so that actual trajectory followed the desired. In addition, in Table 3, the average error of GaPSo control system is the lowest. The P − value for all the joints is lower than 0.05 which shows that the values of the average error for ten times running are different from each other. Table 4 represents analysis of the performance of GaPSo in gait training condition. ME, AE, RMS and R2 represent the maximum error, average error, root-mean-square and coefficient of determination, respectively. ME and AE do not exceed 0.05 which are still in the acceptable range [10]. Furthermore, the maximum of RMS

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M.S. Amiri, R. Ramli and M.F. Ibrahim / Applied Mathematical Modelling 72 (2019) 17–27

(a) Left hip

(c) Left knee

(b) Right hip

(d) Right knee

Fig. 11. Trajectory angle for each joint under gait training condition.

is 0.0170 while, R2 of all joints are more than 99 % which, indicate that the desired nearly fitted to all variability of the actual trajectories. 6. Conclusion This paper presents the method to tune the PID parameters for a four DoF LLE by GaPSo for optimizing the tracking error of the control system. The optimized PID parameters (Kp , Ki , Kd ) were tested in the 3D model of LLE in Gazebo environment for two different conditions of ROM and gait training. The results show that the performance of the control system tuned via GaPSo are better than the GA and PSO. It is because GA set the search space for PSO to reach the global optima. This method can be used in the LLE for rehabilitation by physiotherapists to assist the stroke patients to do lower limb training. However, GaPSo is only suitable for fixed LLE frame. In addition, the influences of noises from mechanical actuators and disturbance of the patient are not considered. Based on these limitations, an adaptive control system based on GaPSo can be extended in the future works to be more robust against real time disturbance and flexible to the adjustable frame of the LLE. Acknowledgment The authors would like to thank Universiti Kebangsaan Malaysia (UKM) and the Ministry of Education Malaysia for financial support received under research grant FRGS/1/2017/TK03/UKM/02/4. References [1] Q. Yan, J. Huang, C. Xiong, Z. Yang, Z. Yang, Data-driven human-robot coordination based walking state monitoring with cane-type robot, IEEE Access 6 (2018) 8896–8908, doi:10.1109/ACCESS.2018.2806563. [2] K. Kim, K. Kim, Progressive treadmill cognitive dual-task gait training on the gait ability in patients with chronic stroke, J. Exerc. Rehabil. 14 (5) (2018) 821–828, doi:10.12965/jer.1836370.185. [3] A. Martinez, B. Lawson, M. Goldfarb, A controller for guiding leg movement during overground walking with a lower limb exoskeleton, IEEE Trans. Robot. 34 (1) (2018) 183–193, doi:10.1109/TRO.2017.2768035. [4] G. Aguirre-Ollinger, J. Edward Colgate, M. A. Peshkin, A. Goswami, Design of an active one-degree-of-freedom lower-limb exoskeleton with inertia compensation, Int. J. Robot. Res. 30 (4) (2011) 486–499, doi:10.1177/0278364910385730.

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